Plot the given function y 2x 3. Plot the function

In the golden age of information technology, few people will buy a graph paper and spend hours drawing a function or an arbitrary set of data, and why do such a chore when you can plot a function online. In addition, it is almost impossible and difficult to calculate millions of expression values ​​for correct display, and despite all efforts, you will get a broken line, not a curve. Therefore, the computer in this case is an indispensable assistant.

What is a function graph

A function is a rule according to which each element of one set is associated with some element of another set, for example, the expression y = 2x + 1 establishes a connection between the sets of all x values ​​and all y values, therefore, this is a function. Accordingly, the graph of the function will be called the set of points whose coordinates satisfy the given expression.

In the figure we see the graph of the function y=x. This is a straight line and each of its points has its own coordinates on the axis X and on the axis Y. Based on the definition, if we substitute the coordinate X some point into this equation, then we get the coordinate of this point on the axis Y.

Services for plotting function graphs online

Consider several popular and best services that allow you to quickly draw a graph of a function.

Opens the list of the most common service that allows you to plot a function graph using an online equation. Umath contains only the necessary tools, such as zooming, moving along the coordinate plane, and viewing the coordinate of the point where the mouse is pointing.

Instruction:

1. Enter your equation in the box after the "=" sign.
2. Click the button "Build Graph".

As you can see, everything is extremely simple and accessible, the syntax for writing complex mathematical functions: with a modulus, trigonometric, exponential - is given right below the graph. Also, if necessary, you can set the equation by the parametric method or build graphs in the polar coordinate system.

Yotx has all the functions of the previous service, but at the same time it contains such interesting innovations as the creation of a function display interval, the ability to build a graph using tabular data, and also display a table with entire solutions.

Instruction:

1. Select the desired schedule method.
2. Enter an equation.
3. Set the interval.
4. Click the button "Build".

For those who are too lazy to figure out how to write down certain functions, this position presents a service with the ability to select the one you need from the list with one click of the mouse.

Instruction:

1. Find the function you need from the list.
2. Click on it with the left mouse button
3. If necessary, enter the coefficients in the field "Function:".
4. Click the button "Build".

In terms of visualization, it is possible to change the color of the graph, as well as hide it or delete it altogether.

Desmos is by far the most sophisticated service for building equations online. By moving the cursor with the left mouse button held down on the graph, you can see in detail all the solutions of the equation with an accuracy of 0.001. The built-in keyboard allows you to quickly write degrees and fractions. The most important plus is the ability to write the equation in any state, without leading to the form: y = f(x).

Instruction:

1. In the left column, right-click on a free line.
2. In the lower left corner, click on the keyboard icon.
3. On the panel that appears, type the desired equation (to write the names of the functions, go to the "A B C" section).
4. The graph is built in real time.

The visualization is just perfect, adaptive, it is clear that the designers worked on the application. Of the pluses, one can note a huge abundance of opportunities, for the development of which you can see examples in the menu in the upper left corner.

There are a lot of sites for plotting functions, but everyone is free to choose for themselves based on the required functionality and personal preferences. The list of the best has been compiled to meet the requirements of any mathematician, young and old. Good luck to you in understanding the "queen of sciences"!

We choose a rectangular coordinate system on the plane and plot the values ​​of the argument on the abscissa axis X, and on the y-axis - the values ​​of the function y = f(x).

Function Graph y = f(x) the set of all points is called, for which the abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y \u003d f (x) is the set of all points in the plane, the coordinates X, at which satisfy the relation y = f(x).

On fig. 45 and 46 are graphs of functions y = 2x + 1 And y \u003d x 2 - 2x.

Strictly speaking, one should distinguish between the graph of a function (the exact mathematical definition of which was given above) and the drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not of the entire graph, but only of its part located in the final parts of the plane). In what follows, however, we will usually refer to "chart" rather than "chart sketch".

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the scope of the function y = f(x), then to find the number f(a)(i.e. the function values ​​at the point x = a) should do so. Need through a dot with an abscissa x = a draw a straight line parallel to the y-axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will be, by virtue of the definition of the graph, equal to f(a)(Fig. 47).

For example, for the function f(x) = x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.

A function graph visually illustrates the behavior and properties of a function. For example, from a consideration of Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values ​​when X< 0 and at x > 2, negative - at 0< x < 2; наименьшее значение функция y \u003d x 2 - 2x accepts at x = 1.

To plot a function f(x) you need to find all points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases, this is impossible, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument X give a finite number of values ​​- say, x 1 , x 2 , x 3 ,..., x k and make a table that includes the selected values ​​of the function.

The table looks like this:

Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).

However, it should be noted that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the marked points and its behavior outside the segment between the extreme points taken remains unknown.

Example 1. To plot a function y = f(x) someone compiled a table of argument and function values:

The corresponding five points are shown in Fig. 48.

Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.

To substantiate our assertion, consider the function

.

Calculations show that the values ​​of this function at points -2, -1, 0, 1, 2 are just described by the above table. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinx; its meanings are also described in the table above.

These examples show that in its "pure" form, the multi-point plotting method is unreliable. Therefore, to plot a given function, as a rule, proceed as follows. First, the properties of this function are studied, with the help of which it is possible to construct a sketch of the graph. Then, by calculating the values ​​of the function at several points (the choice of which depends on the set properties of the function), the corresponding points of the graph are found. And, finally, a curve is drawn through the constructed points using the properties of this function.

We will consider some (the most simple and frequently used) properties of functions used to find a sketch of a graph later, but now we will analyze some commonly used methods for plotting graphs.

Graph of the function y = |f(x)|.

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Recall how this is done. By definition of the absolute value of a number, one can write

This means that the graph of the function y=|f(x)| can be obtained from the graph, functions y = f(x) as follows: all points of the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x), having negative coordinates, one should construct the corresponding points of the graph of the function y = -f(x)(i.e. part of the function graph
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).

Example 2 Plot a function y = |x|.

We take the graph of the function y = x(Fig. 50, a) and part of this graph with X< 0 (lying under the axis X) is symmetrically reflected about the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).

Example 3. Plot a function y = |x 2 - 2x|.

First we plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upwards, the top of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore this part of the graph reflect symmetrically about the x-axis. Figure 51 shows a graph of the function y \u003d |x 2 -2x |, based on the graph of the function y = x 2 - 2x

Graph of the function y = f(x) + g(x)

Consider the problem of plotting the function y = f(x) + g(x). if graphs of functions are given y = f(x) And y = g(x).

Note that the domain of the function y = |f(x) + g(х)| is the set of all those values ​​of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, the functions f(x) and g(x).

Let the points (x 0, y 1) And (x 0, y 2) respectively belong to the function graphs y = f(x) And y = g(x), i.e. y 1 \u003d f (x 0), y 2 \u003d g (x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1+y2),. and any point of the graph of the function y = f(x) + g(x) can be obtained in this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). And y = g(x) by replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), where y 2 = g(x n), i.e., by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 \u003d g (x n). In this case, only such points are considered. X n for which both functions are defined y = f(x) And y = g(x).

This method of plotting a function graph y = f(x) + g(x) is called the addition of graphs of functions y = f(x) And y = g(x)

Example 4. In the figure, by the method of adding graphs, a graph of the function is constructed
y = x + sinx.

When plotting a function y = x + sinx we assumed that f(x) = x, but g(x) = sinx. To build a function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx we will calculate at the selected points and place the results in the table.

The construction of graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember several algorithms for solving such problems, and you can easily plot even the most seemingly complex function. Let's see what these algorithms are.

1. Plotting the function y = |f(x)|

Note that the set of function values ​​y = |f(x)| : y ≥ 0. Thus, the graphs of such functions are always located completely in the upper half-plane.

Plotting the function y = |f(x)| consists of the following simple four steps.

1) Construct carefully and carefully the graph of the function y = f(x).

2) Leave unchanged all points of the graph that are above or on the 0x axis.

3) The part of the graph that lies below the 0x axis, display symmetrically about the 0x axis.

Example 1. Draw a graph of the function y = |x 2 - 4x + 3|

1) We build a graph of the function y \u003d x 2 - 4x + 3. It is obvious that the graph of this function is a parabola. Let's find the coordinates of all points of intersection of the parabola with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 - 4x + 3 = 0.

x 1 = 3, x 2 = 1.

Therefore, the parabola intersects the 0x axis at points (3, 0) and (1, 0).

y \u003d 0 2 - 4 0 + 3 \u003d 3.

Therefore, the parabola intersects the 0y axis at the point (0, 3).

Parabola vertex coordinates:

x in \u003d - (-4/2) \u003d 2, y in \u003d 2 2 - 4 2 + 3 \u003d -1.

Therefore, the point (2, -1) is the vertex of this parabola.

Draw a parabola using the received data (Fig. 1)

2) The part of the graph lying below the 0x axis is displayed symmetrically with respect to the 0x axis.

3) We get the graph of the original function ( rice. 2, shown by dotted line).

2. Plotting the function y = f(|x|)

Note that functions of the form y = f(|x|) are even:

y(-x) = f(|-x|) = f(|x|) = y(x). This means that the graphs of such functions are symmetrical about the 0y axis.

Plotting the function y = f(|x|) consists of the following simple chain of actions.

1) Plot the function y = f(x).

2) Leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the part of the graph specified in paragraph (2) symmetrically to the 0y axis.

4) As the final graph, select the union of the curves obtained in paragraphs (2) and (3).

Example 2. Draw a graph of the function y = x 2 – 4 · |x| + 3

Since x 2 = |x| 2 , then the original function can be rewritten as follows: y = |x| 2 – 4 · |x| + 3. And now we can apply the algorithm proposed above.

1) We build carefully and carefully the graph of the function y \u003d x 2 - 4 x + 3 (see also rice. one).

2) We leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the right side of the graph symmetrically to the 0y axis.

(Fig. 3).

Example 3. Draw a graph of the function y = log 2 |x|

We apply the scheme given above.

1) We plot the function y = log 2 x (Fig. 4).

3. Plotting the function y = |f(|x|)|

Note that functions of the form y = |f(|x|)| are also even. Indeed, y(-x) = y = |f(|-x|)| = y = |f(|x|)| = y(x), and therefore, their graphs are symmetrical about the 0y axis. The set of values ​​of such functions: y ≥ 0. Hence, the graphs of such functions are located completely in the upper half-plane.

To plot the function y = |f(|x|)|, you need to:

1) Construct a neat graph of the function y = f(|x|).

2) Leave unchanged the part of the graph that is above or on the 0x axis.

3) The part of the graph located below the 0x axis should be displayed symmetrically with respect to the 0x axis.

4) As the final graph, select the union of the curves obtained in paragraphs (2) and (3).

Example 4. Draw a graph of the function y = |-x 2 + 2|x| – 1|.

1) Note that x 2 = |x| 2. Hence, instead of the original function y = -x 2 + 2|x| - one

you can use the function y = -|x| 2 + 2|x| – 1, since their graphs are the same.

We build a graph y = -|x| 2 + 2|x| – 1. For this, we use algorithm 2.

a) We plot the function y \u003d -x 2 + 2x - 1 (Fig. 6).

b) We leave that part of the graph, which is located in the right half-plane.

c) Display the resulting part of the graph symmetrically to the 0y axis.

d) The resulting graph is shown in the figure with a dotted line (Fig. 7).

2) There are no points above the 0x axis, we leave the points on the 0x axis unchanged.

3) The part of the graph located below the 0x axis is displayed symmetrically with respect to 0x.

4) The resulting graph is shown in the figure by a dotted line (Fig. 8).

Example 5. Plot the function y = |(2|x| – 4) / (|x| + 3)|

1) First you need to plot the function y = (2|x| – 4) / (|x| + 3). To do this, we return to algorithm 2.

a) Carefully plot the function y = (2x – 4) / (x + 3) (Fig. 9).

Note that this function is linear-fractional and its graph is a hyperbola. To build a curve, you first need to find the asymptotes of the graph. Horizontal - y \u003d 2/1 (the ratio of the coefficients at x in the numerator and denominator of a fraction), vertical - x \u003d -3.

2) The part of the chart that is above or on the 0x axis will be left unchanged.

3) The part of the chart located below the 0x axis will be displayed symmetrically with respect to 0x.

4) The final graph is shown in the figure (Fig. 11).

site, with full or partial copying of the material, a link to the source is required.

It is not difficult to find calculators on the Internet for plotting a graph of a function, which are offered to your attention in this review.

http://www.yotx.ru/

This service can build:

• regular graphs (like y = f(x)),
• given parametrically,
• dot charts,
• graphs of functions in the polar coordinate system.

This is an online service one step:

• Enter the function to be built

In addition to plotting a function graph, you will receive the result of a function study.

Plotting functions:

http://matematikam.ru/calculate-online/grafik.php

You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the chart window, you can hide both the left column and the virtual keyboard.

Benefits of online charting:

• Visual display of introduced functions
• Building very complex graphs
• Plotting implicitly defined graphs (e.g. ellipse x^2/9+y^2/16=1)
• The ability to save charts and get a link to them, which becomes available to everyone on the Internet
• Scale control, line color
• The ability to plot graphs by points, the use of constants
• Construction of several graphs of functions at the same time
• Plotting in polar coordinates (use r and θ(\theta))

The service is in demand for finding intersection points of functions, for displaying graphs for their further transfer to a Word document as illustrations for solving problems, for analyzing the behavioral features of function graphs. The best browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.

http://graph.reshish.ru/

You can build an interactive function graph online. Thanks to this, the graph can be scaled, as well as moved along the coordinate plane, which will allow you not only to get a general idea about the construction of this graph, but also to study in more detail the behavior of the function graph on sections.

To build a graph, select the function you need (on the left) and click on it, or enter it yourself in the input field, and click 'Build'. The variable 'x' is used as an argument.

To set a function nth root from 'x' use the notation x^(1/n) - pay attention to the brackets: without them, following the mathematical logic, you will get (x^1)/n.

You can omit the multiplication sign in expressions with a number: 5x, 10sin(x), 3(x-1); between brackets:(x-7)(4+x); and also between the variable and brackets: x(x-3). Expressions like xsin(x) or xx will throw an error.

Consider the priority of operations and if you are not sure what will be executed first, put extra parentheses. For example: -x^2 and (-x)^2 are not the same.

Keep in mind that the graph may not draw if it tends to infinity in 'y' fast enough, due to the computer's inability to infinitely approach the asymptote in 'x'. This does not mean that the graph breaks off and does not continue to infinity.

In trigonometric functions, the radian measure of the angle is used by default.

http://easyto.me/services/graphic/

In order to build multiple graphs in the same coordinate system, check the box "Build in the same coordinate system" and plot the graphs of the functions one by one.

The service allows you to build graphs of functions in which there are parameters.

For this:

1. Enter a function with parameters and click "Plot"
2. In the window that appears, select which of the variables to build a graph with respect to. Usually this is x.
3. Change the parameter values ​​in the History menu. The schedule will change before your eyes.

http://allcalc.ru/node/650

The service allows you to build graphs of functions in a rectangular coordinate system for a given range of values. In one coordinate plane, you can build several graphs of functions at once.
To build a graph of a function, you need to set the area for plotting the graph (for the variable x and the function y) and enter the value of the dependence of the function on the argument. It is possible to build several graphs at the same time, for this it is necessary to separate the functions with a semicolon. Graphs will be built on the same coordinate plane and will differ in color for clarity.

http://function-graph.ru/

To plot a function online, you just need to enter your function in a special field and click somewhere outside it. After that, the graph of the introduced function will be drawn automatically.

If you need to plot multiple functions at the same time, then click on the blue "Add more" button. After that, another field will open, in which you will need to enter the second function. Her schedule will also be built automatically.

You can adjust the color of the graph lines by clicking on the box located to the right of the function input field. The rest of the settings are right above the graph area. With their help, you can set the background color, the presence and color of the grid, the presence and color of the axes, as well as the presence and color of the numbering of chart segments. If necessary, you can scale the graph of the function using the mouse wheel or special icons in the lower right corner of the drawing area.

After plotting the graph and making the necessary changes to the settings, you can download chart using the big green "Download" button at the very bottom. You will be prompted to save the graph of the function as a PNG image.

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